Create theory section and update tutorials

docs/conf.py

@@ -45,6 +45,7 @@
     "numpydoc",
     "nbsphinx",
     "sphinx_gallery.gen_gallery",
+    "sphinxcontrib.bibtex"
 ]
 
 # Autosummary pages will be generated by sphinx-autogen instead of sphinx-build

docs/content/finite_volume_index.rst

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+.. _finite_volume_index:
+
+Intoduction to Finite Volume
+****************************
+
+What is Finite Volume?
+----------------------
+
+The finite volume method is a method for numerically approximating the solution to partial differential equations.
+Implementation of the finite volume method requires the discretization of continuous functions and variables.
+Discrete representations of functions and variables are organized on a numerical grid (or mesh).
+The final product of the approach is a linear system of equations :math:`\boldsymbol{A \phi=q}`
+that can be solved to compute the discrete approximation of a desired quantity.
+
+.. figure:: ../images/finitevolumeschematic.png
+   :width: 700
+   :align: center
+
+   Conceptual illustrating for solving PDEs with the finite volume method.
+
+In *discretize*, we use a staggered mimetic finite volume approach (:cite:`haber2014,HymanShashkov1999`).
+This approach requires the definitions of variables at either cell-centers, nodes, faces, or edges.
+This method is different from finite difference methods,
+as the final linear system is constructed by approximating the inner products between
+test functions and partial differential equations.
+
+**Contents:**
+
+	- :ref:`Meshes <meshes_index>`
+	- :ref:`Interpolation, Averaging and Differential Operators <operators_index>`
+	- :ref:`Inner Products <inner_products_index>`
+	- :ref:`Discretizing PDEs Derivation Examples <derivation_examples_index>`
+
+**Tutorials and Examples Gallery:**
+
+  - :ref:`Mesh Generation <sphx_glr_tutorials_mesh_generation>`
+  - :ref:`Interpolation, Averaging and Differential Operators <sphx_glr_tutorials_operators>`
+  - :ref:`Inner Products <sphx_glr_tutorials_inner_products>`
+  - :ref:`Discretizing PDEs Derivation Examples <sphx_glr_tutorials_pde>`
+  - :ref:`Examples Gallery <sphx_glr_examples>`
+
+
+Examples
+--------
+
+Below are several examples of the final linear system obtained using the finite volume approach.
+A comprehensive derivation of the final result is not provided here. The full derivations are
+provide in the :ref:`discretizing PDEs derivation examples <derivation_examples_index>` theory section.
+
+Direct Current Resistivity
+^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+The governing equation for the direct current resistivity problem is given by:
+
+.. math::
+	\nabla \cdot \sigma \nabla \phi = -q_s
+
+where
+
+	- :math:`\phi` is the electric potential
+	- :math:`\sigma` is the electrical conductivity within the domain
+	- :math:`q_s` is a general representation of the source term
+	- :math:`\nabla` is the gradient operator
+	- :math:`\nabla \cdot` is the divergence operator
+
+If we choose to define the discrete representation of the electric potential on the nodes,
+the solution for the electric potentials after applying the finite volume approach is given by:
+
+.. math::
+	\boldsymbol{[G^T \! M_{\sigma e} G ]} \boldsymbol{\phi} = \mathbf{q_s}
+
+where :math:`\boldsymbol{G^T \! M_{\sigma e} G }` is a sparse matrix and
+
+	- :math:`\boldsymbol{\phi}` is the discrete approximation to the electric potentials on the nodes
+	- :math:`\boldsymbol{G}` is the :ref:`discrete gradient operator <operators_differential_gradient>`
+	- :math:`\boldsymbol{M_{\sigma e}}` is the :ref:`mass matrix for electrical conductivity <inner_products_isotropic_edges>`
+	- :math:`\boldsymbol{q_s}` is the discrete representation of the source term on the nodes
+
+
+Frequency Domain Electromagnetics
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+The governing equations for the frequency domain electromagnetic problems,
+for a source current, can be expressed using Maxwell's equations:
+
+.. math::
+	\begin{align}
+	&\nabla \times \mu^{-1} \vec{B} - \sigma \vec{E} = \vec{J}_s \\
+	&\nabla \times \vec{E} = - i\omega \vec{B}
+	\end{align}
+
+where
+
+	- :math:`\vec{E}` is the electric field
+	- :math:`\vec{B}` is the magnetic flux density
+	- :math:`\vec{J}_s` is a general representation of the source term
+	- :math:`\sigma` is the electrical conductivity within the domain
+	- :math:`\mu` is the magnetic permeability within the domain
+	- :math:`\omega` is the angular frequency
+	- :math:`\nabla \times` is the curl operator
+
+Here we choose to define the discrete representation of the electric field on edges
+and the discrete representation of the magnetic flux density on faces.
+The solution for the electric potentials after applying the finite volume approach is given by:
+
+.. math::
+	\begin{align}
+	\boldsymbol{C^T \! M_{\mu f} \, b } - \boldsymbol{M_{\sigma e} \, e} = \mathbf{j_s} \\
+	\mathbf{C \, e} = -i \omega \mathbf{b}
+	\end{align}
+
+which can be combined to form a single linear system:
+
+.. math::
+	\boldsymbol{[C^T \! M_{\mu f} C } + i\omega \boldsymbol{M_{\sigma e}]} \mathbf{e} = -i \omega \mathbf{j_s}
+
+where :math:`\boldsymbol{C^T \! M_{\mu f} C } + i\omega \boldsymbol{M_{\sigma e}}` is a sparse matrix and
+
+	- :math:`\boldsymbol{e}` is the discrete approximation to the electric field on edges
+	- :math:`\boldsymbol{b}` is the discrete approximation to the magnetic flux density on faces
+	- :math:`\boldsymbol{C}` is the :ref:`discrete curl operator <operators_differential_curl>`
+	- :math:`\boldsymbol{M_{\sigma e}}` is the :ref:`mass matrix for electrical conductivity <inner_products_isotropic_edges>`
+	- :math:`\boldsymbol{M_{\mu f}}` is the :ref:`mass matrix for the inverse of the magnetic permeability <inner_products_isotropic_reciprocal>`
+	- :math:`\boldsymbol{j_s}` is the discrete representation of the source current density on the edges
+